Multi-structural (MS) games (Immerman 1981, Fagin et al 2021) are
two-player games played using colored pebbles on two sets *A* and
*B* of structures, by two idealized players called Spoiler and
Duplicator. It is known that these games capture the number of
quantifiers, in the sense that Spoiler wins the *r*-round game if
and only if *A* and *B* are distinguishable by an
*r*-quantifier FO sentence. However, if we
**restrict** the number of variables we are allowed to use
in these sentences, MS games or their natural variants no longer capture
the number of quantifiers. In this talk, we develop a game called the
quantifier-variable tree (QVT) game that precisely captures the number
of quantifiers, given a fixed number *k* of variables.